The degree centrality of a node, defined as the number of nodes adjacent to it, is often used as a measure of importance of a node to the structure of a network. This metric can be extended to paths in a network, where the degree centrality of a path is defined as the number of nodes adjacent to it. In this paper, we reconsider the problem of finding the most degree-central shortest path in an unweighted network. We propose a polynomial algorithm with the worst-case running time of $O(|E||V|^2\Delta(G))$, where $|V|$ is the number of vertices in the network, $|E|$ is the number of edges in the network, and $\Delta(G)$ is the maximum degree of the graph. We conduct a numerical study of our algorithm on synthetic and real-world networks and compare our results to the existing literature. In addition, we show that the same problem is NP-hard when a weighted graph is considered. Furthermore, we consider other centrality measures, such as the betweenness and closeness centrality, showing that the problem of finding the most betweenness-central shortest path is solvable in polynomial time and finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not.

翻译：节点的度中心性定义为与其相邻的节点数量，常被用于衡量节点对网络结构的重要性。该指标可扩展至网络中的路径，其中路径的度中心性定义为与其相邻的节点数量。本文重新审视在无权网络中寻找度中心性最大的最短路径问题。我们提出一种多项式算法，其最坏情况运行时间为$O(|E||V|^2\Delta(G))$，其中$|V|$为网络中的顶点数，$|E|$为网络中的边数，$\Delta(G)$为图的最大度。我们在合成网络与真实网络上对该算法进行数值研究，并将结果与现有文献进行对比。此外，我们证明当考虑加权图时，同一问题为NP难问题。进一步地，我们考虑其他中心性度量指标（如介数中心性与接近中心性），证明寻找介数中心性最大的最短路径问题可在多项式时间内求解，而寻找接近中心性最大的最短路径问题无论图是否加权均为NP难问题。